As you start the motion, notice how vector \(\vec{v}\)
remains perpendicular to vector \( \vec{r} \), at all times. This is because
the magnitude of \( \vec{r} \) remains constant, and only its direction
changes. Thus the vector \( d\vec{r} \) which describes the change in
\( \vec{r} \) ends up tangential to the circular trajectory (the
\( \vec{r} \)-circle) in the limit of small angular increments.

The situation repeats itself for vector \( \vec{a} \) which at all times
remains perpendicular to the vector \( \vec{v} \)
(and along \( d \vec{v} \), in the limit of small angular increments).

As a result, the vector \( \vec{a} \) is always in the direction
opposite to that of the vector \( \vec{r} \), i.e. is directed
toward the center of the circle; hence this acceleration is called
centripetal.